Golden Rule transition probability expression

How important, though, is nuclear tunnelling for thermal outer-sphere reactions at ordinary temperature If we work in the Golden Rule formalism, an approximate answer was given some time ago. In harmonic approximation, one obtains from consideration of the Laplace transform of the transition probability (neglecting maximization of pre-exponential terms) the following expressions for free energy (AG ) and enthalpy (AH ) of... 

Electron transfer reactions have also been treated from the quantum mechanical point of view in formal analogy to radiationless transitions, considering the weakly interacting states of a supermolecule AB the probability (rate constant) of the electron transfer is given by a golden rule expression of the type17... 

Those who wish to extend their skills in this direction here should investigate I crmi s golden rule, a general quantum mechanical expression for transition probability. It runs... 

The starting point for all calculations of transition probabilities is the well-known formula (22) sometimes called the Golden Rule. It expresses the transition probability per unit time A in terms of the density of final states... 

With this understanding, we can continue in two ways. First we can use the interaction (13.13) in the golden-rule rate expression—approach we take in Section 13.4.4. Alternatively, we may use the arguments that (1) transitions between states of the high-frequency impurity oscillator can occur with appreciable probability only during close encounters with a bath atom (see footnote 3), and (2) during such encounters, the interactions of the oscillators with other bath atoms is relatively small and can be disregarded, in order to view such encounters as binary collision events. This approach is explored in the next section. 

Thus, to have a proper estimate of photocurrent from expression (75), it is desirable to calculate the transition probability, T E,hv)y using the time-dependent perturbation theory. The general expression of T E, hv) in terms of Fermi s golden rule of time-dependent perturbation theory... 

The transition probability due to an interaction H can be expressed by the Fermi Golden Rule as... 

This expression coincides with the well-known Golden Rule formulation [1] of the molecular transition probability under the external perturbation W= - iEo. 

This last expression is known as Fermi s golden rule. For transitions from one single-particle state to another single-particle state in which case neither the density of states g(e/) nor the dependence of the transition probability on c/ enter, the transition rate takes the form... 

Equation (1.5) is the simplest of all Golden Rule expressions because both the electronic matrix element and the Franck-Condon factor are taken as averages over all interacting vibronic states. A better model employs a Franck-Condon weighted density of states, Pf(F), in order to account for the fact that not all states in the dense manifold couple with i> with the same probability. In any case, the variation in the Franck-Condon factors with electronic energy gap, AE = Ej - Ef, determines the relative rates of radiationless transitions in compounds that contain the same chromophores and hence exhibit similar values of Ujf. The relative magnitudes of the Franck-Condon factors for different vibrational modes also determines the nature of the accepting modes populated preferentially by the radiationless transition. 

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